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Axisymmetric vibration of piezolaminated-CFRP multilayer solid cylinder.


Piezoelectric (PE) materials develop an induced electrical polarization in response to mechanical stress, so vibrating, compressing, or flexing a PE material generates electrical energy. PE materials have been used in actuators, resonators, and transducers for many years. More recent research explores the use of PE materials for applications such as a transformer for a mobile phone battery charger (1), micro generator for embedded micro-systems such as medical implants or building sensors (2-3) and shoe-mounted devices such as a radio tag (4). The main issues for PE devices are the amount of power that can be delivered, the magnitude, and frequency of vibration required to develop sufficient power.

Carbon fiber reinforced plastic (CFRP or CRP), is a very strong composite material or fiber reinforced plastic. Similar to glass-reinforced plastic, which is sometimes simply called fiberglass, the composite material is commonly referred to by the name of its reinforcing fibers (carbon fiber). The plastic is most often epoxy, but other plastics, such as polyester, vinyl ester or nylon, are also sometimes used. Some composites contain both carbon fiber and other fibers such as Kevlar, aluminum and fiberglass reinforcement

Carbon fiber reinforced plastics (CFRP) are widely used as structural materials in applications such as spacecraft structures. However, although they have a high stiffness and low density, these materials are considered to be unsuitable for low-vibration structures because of their inherent low vibration damping (loss factor [eta]=0.001-0.01). The passive damping of CFRP cantilever beams using resistively shunted, surface-bonded piezoelectric ceramic, PbZr[O.sub.3]-PbTi[O.sub.3] (PZT) sheets. The vibration damping of the CFRP beam is significantly improved at the optimal shunting resistance which can be predicted theoretically by (5)

Damage detection and vibration control of a new smart board designed by mounting piezoelectric fibers with metal cores on the surface of a CFRP composite is studied by Takagikiyoshi et al (6). Smart materials based on carbon fiber-reinforced plastics with embedded PZT sensors and actuators are expected to be a favorite composite for vibration damping and noise reduction. Gerhardmook et al (7) has discussed about the imaging techniques using the active piezoceramics as transmitters of acoustic, electromagnetic and thermal fields.

Paul and Nelson (8, 9, 10 and 11) have studied the free vibration of piezocomposite plate and cylinders by embedding LEMV layer between piezoelectric layers. A general frequency equation is derived for axisymmetric vibration of an infinite laminated inner solid and outer hollow cylinder. The outer surface is traction free and connected with electrodes and that are shorted. Numerical calculations are carried out for PZT4/ CFRP /PZT4 /CFRP/PZT4. The attenuation effect is considered through the imaginary part of the dimensionless complex frequency (12).

Fundamental Equations and Method of Analysis

The cylindrical polar coordinate system (r, [theta],z) is used for composite piezoelectric cylinder. The superscripts l = 1, 3,5 are taken to denote the inner solid, middle and outer hollow piezoelectric cylinders respectively.

The governing equations for hexagonal (6 mm) class are Paul and Nelson (8).


Here [u.sup.l], [w.sup.l] are the displacement components along r, z directions

[[phi].sup.l] the potentials and

[c.sup.l.sub.ij]--Elastic constants

[e.sup.l.sub.ij]- Piezoelectric constants

[[epsilon].sup.l.sub.ij]- Dielectric constants

[[rho].sup.l.sub.ij] - Density of the materials

The comma followed by superscripts denotes the partial differentiation with respect to those variables and t is the time.

The solution of equation (1.1) is taken in the form.

[u.sup.l] = [u.sup.l.sub.,r] exp i(kz pt)

[w.sup.l] = (i / a)[w.sup.l] exp i(kz pt)

[[phi].sup.l] = i([c.sup.l.sub.44]/[ae.sup.l.sub.33])[[phi].sup.l] i(kz pt) (1.2)

where p is the angular frequency, k wave number and 'a' is the inner radius of the cylinder.

Substituting equation (1.2) along with the dimensionless variables x = r/a and [epsilon] = ka (k = [pi]/wave length) in equation (1.1) yields the following equation for the inner and outer cylinder.



Equation (1.3) can be expressed as



The solutions of equation (1.4)


Here [([[alpha].sup.l.sub.j] a).sup.2] are the non-zero roots of


The arbitrary constants [d.sup.l.sub.j] and [e.sup.l.sub.j] are given by


For isotropic materials, the governing equations are


where [[bar.u].sup.l] is the displacement vector

[[lambda].sup.l] [c.sub.12], [[mu].sup.l] = ([c.sub.11] - [c.sub.12])/2 are Lame's constants

[[rho].sup.l] is the mass density and t is the time.

The solution of equation (1.9) is taken as


Using the solution in equation (1.10) and the dimensionless variables x and [member of], equation (1.9) can be simplified as



The equation (1.11) can be written as

([[nabla].sup.4] + [P.sup.l] [[nabla].sup.2] + [Q.sup.l])([u.sup.l], [w.sup.l]) = 0 (1.12)


The solutions of equation (1.12) are


where [([[alpha].sup.l.sub.j]a).sup.2] is the non zero roots of

[([[alpha].sup.l.sub.j]a).sup.4] + [P.sup.l] [([[alpha].sup.l]a).sup.2] - [Q.sup.l] = 0 (1.15)

And the arbitrary constants [d.sup.l.sub.j] are obtained from


Boundary--Interface Conditions and Frequency Equations The frequency equations can be obtained by using the following boundary and interface conditions.

On the traction free outer surface [T.sup.l.sub.rr] = [T.sup.l.sub.rz] = [[phi].sup.l] with l = 5. At the interface between (outer and middle and middle and inner) cylinders [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], with l =1, 2,3,4,5.

The frequency equation is obtained as a 23 x 23 determinant equation by substituting the solutions in the boundary--interface conditions. It is written as

[absolute value of ([D.sub.i,j])] = 0, (i, j = 1, 2 ... 23) (1.17)

and the non-zero elements by varying j from 1 to 3 and k varies from 1 to 2 are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and the following elements are obtained by replacing [J.sub.n] and [J.sub.n+1] by [Y.sub.n] and [Y.sub.n+1].

In the respective expression

D(i, k +5), (i = 1, 2, 3, 4) (k= 1, 2) and

At the second inter face x = x2, non zero elements along the following rows D (i, j), (i = 6,7,8,9,10) (j = 4,5,6.... 13) are obtained on replacing [x.sub.1] by [x.sub.2] and superscript 1 by 2 in order. Similarly, at the third interface x = [x.sub.3], the nonzero elements D (i, j), (i=11, 12, 13,14,15) (j = 8, 9,10.... 17) can be had from the second interface layers by assigning [x.sub.3] for [x.sub.2] .The non-zero elements at the fourth interface are, D (i, j), (i=16,17,18,19,20) (j = 14,15,16.... 23) can be obtained by assigning [x.sub.4] for [x.sub.3]. The non zero elements at the fifth layer are, D (k, j), ((i = 21, 22, 23) and j = 18, 19, 20 ... 23) are obtained on replacing [x.sub.5] by [x.sub.5]. The frequency equations derived above are valid for different inner solid, middle and outer hollow materials of 6mm class and arbitrary thickness of layers.

Numerical Results

The frequency equation (1.17) and corresponding equation are numerically evaluated for PZT4/CFRP/PZT4/CFRP/PZT4. Material Constants of CFRP bonding layer are taken from Ashby and Jones (13). The elastic piezoelectric and dielectric constants of PZT4 are taken from Brelincourt et al (14). The roots of the frequency equations are calculated using Muller's method. The complex frequencies for the axisymmetric waves in the first and second modes are given in Tables 1.1 and 1.2. The attenuation in the case of piezocomposite with CFRP (5-layer Model) as the middle core is more when compared to CFRP (3 layer model) piezocomposite with LEMV (when N=0.66) as core material. The dispersion curves for the real part of frequency against the dimensionless wave numbers are plotted for the first and second axisymmetric mode in Figure 2.1 and Figure 2.2. The bold, discontinuous and dotted lines indicate the dispersion curves in the axisymmetric vibrations of the piezolaminated-CFRP (5-Layer Model), piezolaminated-CFRP (3-Layer Model) and piezolaminated-LEMV (with N=0.66) cylinders.




The frequency equation for free axisymmetric vibration of piezolaminated Multilayer solid cylinder with isotropic CFRP bonding layers is derived. The numerical results are carried out for PZT4/ CFRP/ PZT4/ CFRP/ PZT4 and are compared with piezolaminated-CFRP Multilayer (3-layer) solid cylinder and piezolaminated-LEMV (With N=0.66) cylinder. It is observed from the numerical data that the attenuation effect in the present model with CFRP bonding layers is high when compared to the piezolaminated-LEMV (With N=0.66) cylinders. Also the damping effect in the present five layered model is high when compared with three layered solid and hollow piezocomposite models, the present smart material model may find its real time applications for vibration damping and noise reduction devices.


[1] Navas, J Bove. T, Cobos. J.A, Nuno. F, and Brebol. K, (2001) "Miniaturized Battery Charger Using Piezoelectric Transformers," Proc. 6th Annual IEEE Applied Power Electronics Conf., vol. 1, pp. 492-96.

[2] Glynne-Jones. P, Beeby. S.P, and White. N.M, (2001) "Towards a Piezoelectric Vibration-Powered Micro-generator," Proc. IEEE, vol. 148, pp. 68-72.

[3] Williams C.B. and Yates. R.B, (1996) "Analysis of a Microelectric Generator for Microsystems," Sensors and Actuators A, vol. A52, No. 1-3, pp. 8-11.

[4] Shenck N.S. and Paradiso. J.A, (2001) "Energy Scavenging with ShoeMounted Piezo-electrics," IEEE Micro., vol. 21, pp. 30-42.

[5] Toshio Tanimoto et. al, (1997) "Passive Damping Performance of an Adaptive Carbon-Fiber Reinforced Plastics/Lead Zirconate Titanate Beam", Jpn. J. Appl. Phys. vol. 36, pp. 6110-6113.

[6] Takagikiyoshi, Sato Hiroshi and Saigo Muneharu (2006) "Vibration control of a smart structure embedded with metal-core piezoelectric fibers", Advanced Composite Materials, Vol. 15. pp. 403-417.

[7] Gerhardmook, Juergen Pohl, Fritz Michel and Andreas Hilling (2003) "From non-destructive inspection to health monitoring of smart cfrp-composites", 8th ECNDT, Vol. 8, No. 2, Barcelona.

[8] Paul H.S. and Nelson V.K. (1994) "Axisymmetric vibration of piezoelectric composite cylinders", Proc. Third international congress on Air and Structureborne Sound and Vibration, Vol.1, pp.137-144.

[9] Paul H.S. and Nelson V.K. (1995) "Wave propagation in piezocomposite plates", Proc. Indian Nat. Sci. Acad., pt-A, pp.221-228.

[10] Paul H.S. and Nelson V.K. (1996) "Axisymmetric vibration of piezocomposite hollow circular cylinder", Acta Mechanica, Vol. 116, pp. 213-222.

[11] Paul H.S. and Nelson V.K. (1996a) "Flexural vibration of piezocomposite hollow cylinder", J. Acoust. Soc. Am., Vol.99, pp. 309-313.

[12] Sinha B.K., Plona T.J., Kostek S. and Chang S.K. (1992) "Axisymmetric wave propagation in fluid-loaded cylindrical shells. Part-I: Theory, part-II: Theory vesus experiment", J, Acoust. Soc. Am, Vol. 92, pp. 1132-1155.

[13] Ashby M.F. and Jones D.R.H. (1986) "Engineering Materials 2", Pergamon Press, London.

[14] Brelincourt D.A. Curran D.R. and Jafee H (1964) "Physical acoustics", Vol.1, Academic Press, New York.

Nehru E.S. (1) and Dhandapani. A. (2)

(1) Department of Mathematics, IBRI College of Technology, IBRI, Sultanate OMAN,

(2) Department of Mathematics, JA Institute of Engineering and Technology, Chennai
Table 1.1 : Different value of complex frequencies for real wave
numbers in the first axial mode of piezocomposite solid cylinder

Wave no                          Frequencies

                                 With middle core   With middle core
([epsilon])   With middle core   CFRP               CFRP-5Layer
              LEMV (N=0.66)      (3-Layer Model)    (Present Model)

0.1           0.4966+i 0.0000     0.1335+i 0.7092    0.3649+i 0.3899
0.25          0.8500+i 0.0000     0.4821+i 0.0000    0.4900+i 0.3058
0.5           1.4000+i 0.0000     0.5000+i 0.0000    0.9433+i 0.0056
0.75          1.6513+i 0.2145     1.1767+i 0.0006    1.6053+i 0.0051
1.0           2.1997+i 0.0000     1.2901+i 1.0719    1.7156+i 0.0087
1.25          2.5672+i 0.0004     1.3818+i 0.0048    1.8523+i 0.7897
1.5           2.6500+i 0.0000     1.4448+i 0.3663    2.2503+i 0.0957
1.75          2.6937+i 0.0000     1.8037+i 0.0001    2.6315+i 1.1298
2.0           3.0500+i 0.0000     1.8624+i 0.0039    3.0148+i 1.2857
2.25          3.4449+i 0.0000     2.1428+i 0.0050    3.3923+i 1.4471
2.5           3.5500+i 0.0000     2.3302+i 0.0082    3.9419+i 0.7222
2.75          3.8241+i 0.0022     2.7650+i 0.4003    4.0610+i 0.2844
3.0           4.0400+i 0.0010     2.8898+i 0.0006    4.5303+i 1.9547

Table 1.2 : Different value of complex frequencies for real wave
numbers in the second axial mode of piezocomposite solid cylinder

Wave no                          Frequencies

([epsilon])   With middle core   With mid core     With middle core
              LEMV (N=0.66)      CFRP              CFRP-5Layer
                                 (3-Layer Model    (Present Model)

0.1           0.5500+i 0.0000    0.2042+i 0.0000   0.3945+i 0.1152
0.25          0.9500+i 0.0000    0.6000+i 0.0000   0.7424+i 0.0042
0.5           1.4445+i 0.0000    0.9000+i 0.0000   1.2034+i 0.7270
0.75          2.3000+i 0.0000    1.3999+i 0.0000   1.9166+i 0.0061
1.0           2.6500+i 0.0000    1.5002+i 0.0000   2.2004+i 0.0000
1.25          2.7500+i 0.0000    1.4837+i 0.0013   2.5028+i 0.1229
1.5           3.0375+i 0.0000    1.7000+i 0.0000   2.7502+i 0.1368
1.75          3.3000+i 0.0000    1.8000+i 0.0000   3.1708+i 0.1558
2.0           3.5000+i 0.0000    2.1000+i 0.0000   3.5921+i 0.0020
2.25          3.5719+i 0.0000    2.2710+i 0.0014   4.0752+i 0.0060
2.5           3.6500+i 0.0000    2.4443+i 0.0014   4.6288+i 0.1313
2.75          3.9406+i 0.0013    3.0045+i 0.0020   5.0195+i 0.0018
3.0           4.1500+i 0.0000    3.3000+i 0.0000   5.4833+i 0.0015
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Title Annotation:carbon fiber-reinforced plastics
Author:Nehru, E.S.; Dhandapani, A.
Publication:International Journal of Applied Engineering Research
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2008
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